{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2D PSV wave propagation in a homogenous block-model\n",
    "===="
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The propagation of waves in a general elastic medium can be described by a system of coupled linear partial differential equations. They consist of the equations of motion\n",
    "\n",
    "$$\\rm{\\rho \\frac{\\partial v_i}{\\partial t} = \\frac{\\partial \\sigma_{ij}}{\\partial x_j} + fs_i}$$\n",
    "   \n",
    "which simply state that the momentum of the particles in the medium, the product of density $\\rm{\\rho}$ and the displacement velocity $\\rm{v_i}$, can be changed by surface forces, described by the stress tensor $\\rm{\\sigma_{ij}}$ or body forces $\\rm{fs_i}$. These equations describe a general medium, like gas, fluid, solid or plasma. The material specific properties are introduced by additional equations which describe how the medium reacts when a certain force is applied. In the isotropic elastic case this can be described by a linear stress-strain relationship:  \n",
    "\n",
    "$$\\begin{split}\n",
    "\\rm{\\sigma_{ij}}&\\rm{=\\lambda \\theta \\delta_{ij} + 2 \\mu \\epsilon_{ij}}\\\\\n",
    "\\rm{\\epsilon_{ij}}&\\rm{=\\frac{1}{2}\\biggl(\\frac{\\partial u_i}{\\partial x_j}+\\frac{\\partial u_j}{\\partial x_i}\\biggr)}\n",
    "\\end{split}\n",
    "$$\n",
    "\n",
    "where $\\rm{\\lambda}$ and $\\rm{\\mu}$ are the Lamé parameters, $\\rm{\\epsilon_{ij}}$ the strain tensor, $\\rm{\\theta = \\epsilon_{11} + \\epsilon_{22} + \\epsilon_{33}}$ the cubic dilatation, $\\rm{\\delta_{ij}}$ the Kronecker Delta and $\\rm{u_i}$ the displacement. By taking the time derivative of the stress-strain relationship and the strain tensor, we can derive the following partial differential equations to describe wave propagtion in a general 3D isotropic elastic medium:\n",
    "\n",
    "$$\\begin{split}\n",
    "\\rm{\\rho \\frac{\\partial v_i}{\\partial t}} &\\rm{= \\frac{\\partial \\sigma_{ij}}{\\partial x_j} + fs_i}\\\\\n",
    "\\rm{\\frac{\\partial \\sigma_{ij}}{\\partial t}} &\\rm{= \\lambda \\frac{\\partial \\theta}{\\partial t} \\delta_{ij} + 2 \\mu \\frac{\\partial \\epsilon_{ij}}{\\partial t}}\\\\\n",
    "\\rm{\\frac{\\partial \\epsilon_{ij}}{\\partial t}}&\\rm{=\\frac{1}{2}\\biggl(\\frac{\\partial v_i}{\\partial x_j}+\\frac{\\partial v_j}{\\partial x_i}\\biggr)}\n",
    "\\end{split}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Equations of motion for 2D PSV wave propagation in an isotropic elastic medium\n",
    "----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In case of certain symmetries and model limitations, the general 3D seismic wave propagation in isotropic elastic media can be significantly simplified. Assuming only non-zero particle displacements in the x-y-plane (PSV problem), where x denotes the horizontal distance and y the depth, wave propagation can be described by the following system of partial differential equations:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\rm{\\rho \\frac{\\partial v_x}{\\partial t} = \\frac{\\partial \\sigma_{xx}}{\\partial x} + \\frac{\\partial \\sigma_{xy}}{\\partial y} + fs_x,} $$\n",
    "\n",
    "\n",
    "$$\\rm{\\rho \\frac{\\partial v_y}{\\partial t} = \\frac{\\partial \\sigma_{xy}}{\\partial x} + \\frac{\\partial \\sigma_{yy}}{\\partial y} + fs_y,} $$\n",
    "\n",
    "$$\\rm{\\frac{\\partial \\sigma_{xx}}{\\partial t} = (\\lambda + 2 \\mu) \\frac{\\partial v_{x}}{\\partial x} + \\lambda \\frac{\\partial v_{y}}{\\partial y},} $$\n",
    "\n",
    "$$\\rm{\\frac{\\partial \\sigma_{yy}}{\\partial t} = \\lambda \\frac{\\partial v_{x}}{\\partial x} + (\\lambda + 2 \\mu) \\frac{\\partial v_{y}}{\\partial y},} $$\n",
    "\n",
    "$$\\rm{\\frac{\\partial \\sigma_{xy}}{\\partial t} = \\mu \\biggl(\\frac{\\partial v_{x}}{\\partial y} + \\frac{\\partial v_{y}}{\\partial x}\\biggr),}  $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $\\rm{\\rho}$ is the density, $\\rm{\\lambda}$ and $\\rm{\\mu}$ the Lamé parameters, $\\rm{(v_x,\\; v_y)}$ particle velocity vector, $\\rm{\\sigma_{xx}}$, $\\rm{\\sigma_{yy}}$, $\\rm{\\sigma_{xy}}$ stress tensor components, ($\\rm{fs_x}$, $\\rm{fs_y}$) directed body force vector, respectively."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finite difference discretization on a staggered grid\n",
    "----"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the numerical solution of the elastic equations of motion have to be discretized in time and space on a grid. The particle velocities $\\rm{\\mathbf{v}}$, the stresses $\\rm{\\sigma_{ij}}$, the Lamé parameters $\\rm{\\lambda}$ and $\\rm{\\mu}$ are calculated and defined at discrete Cartesian coordinates $\\rm{x=i\\; dh}$, $\\rm{y=j\\; dh}$ and discrete times $\\rm{t=n\\; dt}$. \n",
    "$\\rm{dh}$ denotes the spatial distance between two adjacent grid points and $\\rm{dt}$ the difference between two successive time steps. Therefore every grid point is located in the interval  $\\rm{i \\in N | [1,nx]}$, $\\rm{j \\in N | [1,ny]}$ and $\\rm{n \\in N | [1,nt]}$, where\n",
    "$\\rm{nx}$, $\\rm{ny}$ and $\\rm{nt}$ are the number of discrete spatial grid points and time steps, respectively."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally the partial derivatives are replaced by **finite-difference (FD)** operators. Two types of operators can be distinguished, forward and backward operators $\\rm{D^+,\\;D^-}$. The derivative of a function f(x) with respect to a variable x can be approximated by the following 2nd order operators:\n",
    "\n",
    "$$\\rm{D^+_x f = \\frac{f_{i+1}-f_{i}}{dh} \\hspace{1 cm} \\text{forward operator}} $$\n",
    "\n",
    "$$\\rm{D^-_x f = \\frac{f_i-f_{i-1}}{dh} \\hspace{1 cm} \\text{backward operator}} $$\n",
    "\n",
    "To calculate the spatial derivatives of the wavefield variables at the correct positions with respect to each other, the variables are not placed on the same grid points, but staggered by half of the spatial grid point distance ([Virieux 1986](https://www.researchgate.net/publication/228078301_P-SV_wave_propagation_in_heterogeneous_media_Velocity-stress_finite-difference_method)). \n",
    "Figure 1 shows the distribution of the material parameters and wavefield variables on the spatial grid. \n",
    "To guarantee the stability of the **staggered grid** code, the Lamé parameter $\\rm{\\mu}$ and density $\\rm{\\rho}$ have to be averaged harmonically and arithmetically ([Moczo et al. 2004](http://www.quest-itn.org/library/training-material/the-finite-difference-method-for-seismologists.-an-introduction/at_download/file), [Bohlen 2006](https://www.researchgate.net/publication/249866080_Accuracy_of_heterogeneous_staggered-grid_finite-difference_modeling_of_Rayleigh_waves)), respectively\n",
    "\n",
    "$$\\rm{\\mu_{xy}(j+1/2,i+1/2)=\\biggl[\\frac{1}{4}\\biggl(\\mu^{-1}_{j,i}+\\mu^{-1}_{j,i+1}+\\mu^{-1}_{j+1,i+1}+\\mu^{-1}_{j+1,i}\\biggr)\\biggr]^{-1}} $$\n",
    "\n",
    "$$\\rm{\\rho_x(j,i+1/2) = 0.5\\, (\\rho_{j,i+1}+\\rho_{j,i})} $$\n",
    "\n",
    "$$\\rm{\\rho_y(j+1/2,i) = 0.5\\, (\\rho_{j+1,i}+\\rho_{j,i})} $$\n",
    "\n",
    "![Staggered grid](./fig/SSG_cart_new_small.png)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Discretized equations of motion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the next step we discretize the equations of motion for the 2D PSV problem using a staggered finite difference approach. First, we discretize the x-compoment of the momentum equation by approximating the spatial derivatives"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\rm{\\frac{\\partial \\sigma_{xx}}{\\partial x} \\approx \\frac{\\sigma_{xx}(j,i+1) - \\sigma_{xx}(j,i)}{dh}}, \\rm{\\frac{\\partial \\sigma_{xy}}{\\partial y} \\approx \\frac{\\sigma_{xy}(j+1/2, i) - \\sigma_{xy}(j-1/2,i)}{dh}} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and the LHS of the x-momentum equation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\rho \\rm{\\frac{\\partial v_x}{\\partial t} \\approx \\rho_x(j,i+1/2) \\frac{v_x^{n+1/2}(j,i+1/2) - v_x^{n-1/2}(j,i+1/2)}{dt}} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Inserting in the partial differential equation "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\rm{\\rho \\frac{\\partial v_x}{\\partial t} = \\frac{\\partial \\sigma_{xx}}{\\partial x} + \\frac{\\partial \\sigma_{xy}}{\\partial y}} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "leads to "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\rho_x(j,i+1/2) \\frac{v_x^{n+1/2}(j,i+1/2) - v_x^{n-1/2}(j,i+1/2)}{dt} = \\frac{\\sigma_{xx}^n(j,i+1) - \\sigma_{xx}^n(j,i)}{dh} + \\frac{\\sigma_{xy}^n(j+1/2, i) - \\sigma_{xy}^n(j-1/2,i)}{dh} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After rearranging for $v_x^{n+1/2}(j,i+1/2)$ we get the following explicit FD scheme for the x-component of the momentum equation:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\rm{v_x^{n+1/2}(j,i+1/2) = v_x^{n-1/2}(j,i+1/2) + \\frac{dt}{dh\\cdot \\rho_x(j,i+1/2)}\\cdot \\biggl(\\sigma^n_{xx}(j,i+1) - \\sigma^n_{xx}\n",
    "(j,i) + \\sigma^n_{xy}(j+1/2, i) - \\sigma^n_{xy}(j-1/2,i) \\biggr)} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using a similar approach we can derive the FD scheme for the y-compoment of the momentum equation ..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\rm{v_y^{n+1/2}(j,i+1/2) = v_y^{n-1/2}(j,i+1/2) + \\frac{dt}{dh\\cdot \\rho_y(j+1/2,i)}\\cdot \\biggl(\\sigma^n_{xy}(j, i+1/2) - \\sigma^n_{xy}(j,i-1/2) + \\sigma^n_{yy}(j+1,i) - \\sigma^n_{yy}(j,i) \\biggr)} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "... and the stress-strain relationship ..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{split}   \n",
    "\\rm{\\sigma^{n+1}_{xx}(j,i)}\\;&\\rm{= \\sigma_{xx}^{n}(j,i) + dt\\cdot\\lambda(j,i)\\cdot \\biggl(v^{n+1/2}_{xx}(j,i) + v^{n+1/2}_{yy}(j,i) \\biggr) + 2 dt\\cdot  \\mu(j,i) \\cdot  v^{n+1/2}_{xx}(j,i)}\\\\ \n",
    "\\rm{\\sigma^{n+1}_{yy}(j,i)}\\;&\\rm{= \\sigma_{yy}^{n}(j,i) + dt\\cdot\\lambda(j,i)\\cdot \\biggl(v^{n+1/2}_{xx}(j,i) + v^{n+1/2}_{yy}(j,i) \\biggr) + 2 dt\\cdot  \\mu(j,i) \\cdot  v^{n+1/2}_{yy}(j,i)}\\\\ \n",
    "\\rm{\\sigma^{n+1}_{xy}(j+1/2,i+1/2)}\\;&\\rm{=\\sigma^{n}_{xy}(j+1/2,i+1/2) + dt\\cdot\\mu_{xy}(j+1/2,i+1/2)\\biggl(v^{n+1/2}_{xy}(j+1/2,i+1/2) + v^{n+1/2}_{yx}(j+1/2,i+1/2)\\biggr)}\\\\\n",
    "\\end{split}\n",
    "$$\n",
    "\n",
    "with the spatial derivatives\n",
    "\n",
    "$$\n",
    "\\begin{split}  \n",
    "\\rm{v_{xx}(j,i)}\\; & \\rm{= \\frac{v_x(j,i+1/2)-v_x(j,i-1/2)}{dh}}\\\\ \n",
    "\\rm{v_{yy}(j,i)}\\; & \\rm{= \\frac{v_y(j+1/2,i)-v_y(j-1/2,i)}{dh}}\\\\ \n",
    "\\rm{v_{yx}(j+1/2,i+1/2)}\\; & \\rm{= \\frac{v_y(j+1/2, i+1)-v_y(j+1/2, i)}{dh}}\\\\ \n",
    "\\rm{v_{xy}(j+1/2,i+1/2)}\\; & \\rm{= \\frac{v_x(j+1,i+1/2)-v_x(j, i+1/2)}{dh}}\\\\\n",
    "\\end{split}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Implementing 2D PSV code\n",
    "----\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# load all necessary libraries \n",
    "import numpy\n",
    "from matplotlib import pyplot, cm\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from numba import jit\n",
    "%matplotlib notebook"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# spatial discretization\n",
    "nx = 601\n",
    "ny = 601\n",
    "dh = 5.0\n",
    "x = numpy.linspace(0, dh*(nx-1), nx)\n",
    "y = numpy.linspace(0, dh*(ny-1), ny)\n",
    "X, Y = numpy.meshgrid(x, y)\n",
    "\n",
    "# time discretization\n",
    "T = 0.55\n",
    "dt = 0.6e-3\n",
    "nt = numpy.floor(T/dt)\n",
    "nt = nt.astype(int)\n",
    "\n",
    "# snapshot frequency [timesteps] \n",
    "isnap = 10\n",
    "\n",
    "# wavefield clip\n",
    "clip = 2.5e-2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# define model parameters\n",
    "rho = 7100.0\n",
    "vp = 2955.0\n",
    "vs = 2362.0\n",
    "mu = rho * vs * vs\n",
    "lam = rho * vp * vp - 2 * mu"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because we are currently dealing with a homogeneous block model, we don't have to care about the artihmetic and harmonic averaging of density and shear modulus, respectively. In the next step we define the FD updates for particle velocity and stresses and assemble the 2D PSV FD code."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Update particle velocities**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "@jit(nopython=True) # use JIT for C-performance\n",
    "def update_v(vx, vy, sxx, syy, sxy, nx, ny, dtdx, rhoi):\n",
    "    \n",
    "    for j in range(1, ny-1):\n",
    "        for i in range(1, nx-1):\n",
    "                \n",
    "            # calculate spatial derivatives    \n",
    "            sxx_x = sxx[j, i+1] - sxx[j, i]\n",
    "            syy_y = syy[j+1, i] - syy[j, i]\n",
    "            sxy_x = sxy[j, i] - sxy[j, i-1]\n",
    "            sxy_y = sxy[j, i] - sxy[j-1, i]        \n",
    "        \n",
    "            # update particle velocities \n",
    "            vx[j, i] = vx[j, i] + dtdx * rhoi * (sxx_x + sxy_y)\n",
    "            vy[j, i] = vy[j, i] + dtdx * rhoi * (sxy_x + syy_y)\n",
    "        \n",
    "    return vx, vy    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Update stresses**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "@jit(nopython=True) # use JIT for C-performance\n",
    "def update_s(vx, vy, sxx, syy, sxy, nx, ny, dtdx, lam, mu):\n",
    "    \n",
    "    for j in range(1, ny-1):\n",
    "        for i in range(1, nx-1):\n",
    "            \n",
    "            # calculate spatial derivatives\n",
    "            vxx = vx[j][i] - vx[j][i-1]\n",
    "            vyy = vy[j][i] - vy[j-1][i]        \n",
    "            vyx = vy[j][i+1] - vy[j][i]\n",
    "            vxy = vx[j+1][i] - vx[j][i]\n",
    "                    \n",
    "            # update stresses\n",
    "            sxx[j, i] = sxx[j, i] + dtdx * ( lam * (vxx + vyy) + 2.0 * mu * vxx )\n",
    "            syy[j, i] = syy[j, i] + dtdx * ( lam * (vxx + vyy) + 2.0 * mu * vyy )\n",
    "            sxy[j, i] = sxy[j, i] + dtdx * (  mu * (vyx + vxy) )\n",
    "        \n",
    "    return sxx, syy, sxy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Assemble the 2D PSV code**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def psv_mod(nt, nx, ny, dt, dh, rho, lam, mu, clip, isnap, X, Y):\n",
    "    \n",
    "    # initialize wavefields\n",
    "    vx = numpy.zeros((ny, nx))\n",
    "    vy = numpy.zeros((ny, nx))\n",
    "    sxx = numpy.zeros((ny, nx)) \n",
    "    syy = numpy.zeros((ny, nx))\n",
    "    sxy = numpy.zeros((ny, nx))\n",
    "    \n",
    "    # define some parameters\n",
    "    dtdx = dt / dh\n",
    "    rhoi = 1.0 / rho\n",
    "    \n",
    "    # define source wavelet parameters\n",
    "    fc = 17.0\n",
    "    tshift = 0.0\n",
    "    ts = 1.0 / fc\n",
    "    \n",
    "    # source position [gridpoints]\n",
    "    jjs = 300\n",
    "    iis = 300\n",
    "    \n",
    "    # initalize animation \n",
    "    fig = pyplot.figure(figsize=(11,7))\n",
    "    extent = [numpy.min(X),numpy.max(X),numpy.min(X),numpy.max(Y)]\n",
    "    image = pyplot.imshow(vy, animated=True, cmap=cm.seismic, interpolation='nearest', vmin=-clip, vmax=clip)\n",
    "    \n",
    "    pyplot.colorbar()\n",
    "    pyplot.title('Wavefield vy')\n",
    "    pyplot.xlabel('X [m]')\n",
    "    pyplot.ylabel('Y [m]')\n",
    "    pyplot.gca().invert_yaxis()\n",
    "\n",
    "    pyplot.ion()\n",
    "    pyplot.show(block=False)\n",
    "        \n",
    "    # loop over timesteps \n",
    "    for n in range(nt):                \n",
    "        \n",
    "        # define Ricker wavelet\n",
    "        t = n * dt\n",
    "        tau = numpy.pi * (t - 1.5 * ts - tshift) / (1.5 * ts)\n",
    "        amp = (1.0 - 4.0 * tau * tau) * numpy.exp(-2.0 * tau * tau)\n",
    "        \n",
    "        # update particle velocities                            \n",
    "        vx, vy = update_v(vx, vy, sxx, syy, sxy, nx, ny, dtdx, rhoi)\n",
    "        \n",
    "        # apply vertical impact source term @ source position\n",
    "        vy[jjs, iis] = vy[jjs, iis] + amp        \n",
    "        \n",
    "        # update stresses                \n",
    "        sxx, syy, sxy = update_s(vx, vy, sxx, syy, sxy, nx, ny, dtdx, lam, mu)\n",
    "        \n",
    "        # display vy snapshots \n",
    "        if (n % isnap) == 0:\n",
    "            image.set_data(vy)\n",
    "            fig.canvas.draw()\n",
    "            \n",
    "    return vx, vy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's run the 2D PSV code for a homogeneous block model:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "        return WebSocket;\n",
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       "        return MozWebSocket;\n",
       "    } else {\n",
       "        alert('Your browser does not have WebSocket support.' +\n",
       "              'Please try Chrome, Safari or Firefox ≥ 6. ' +\n",
       "              'Firefox 4 and 5 are also supported but you ' +\n",
       "              'have to enable WebSockets in about:config.');\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n",
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       "\n",
       "    this.ws = websocket;\n",
       "\n",
       "    this.supports_binary = (this.ws.binaryType != undefined);\n",
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       "    if (!this.supports_binary) {\n",
       "        var warnings = document.getElementById(\"mpl-warnings\");\n",
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       "            warnings.style.display = 'block';\n",
       "            warnings.textContent = (\n",
       "                \"This browser does not support binary websocket messages. \" +\n",
       "                    \"Performance may be slow.\");\n",
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       "\n",
       "    this.imageObj = new Image();\n",
       "\n",
       "    this.context = undefined;\n",
       "    this.message = undefined;\n",
       "    this.canvas = undefined;\n",
       "    this.rubberband_canvas = undefined;\n",
       "    this.rubberband_context = undefined;\n",
       "    this.format_dropdown = undefined;\n",
       "\n",
       "    this.image_mode = 'full';\n",
       "\n",
       "    this.root = $('<div/>');\n",
       "    this._root_extra_style(this.root)\n",
       "    this.root.attr('style', 'display: inline-block');\n",
       "\n",
       "    $(parent_element).append(this.root);\n",
       "\n",
       "    this._init_header(this);\n",
       "    this._init_canvas(this);\n",
       "    this._init_toolbar(this);\n",
       "\n",
       "    var fig = this;\n",
       "\n",
       "    this.waiting = false;\n",
       "\n",
       "    this.ws.onopen =  function () {\n",
       "            fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n",
       "            fig.send_message(\"send_image_mode\", {});\n",
       "            fig.send_message(\"refresh\", {});\n",
       "        }\n",
       "\n",
       "    this.imageObj.onload = function() {\n",
       "            if (fig.image_mode == 'full') {\n",
       "                // Full images could contain transparency (where diff images\n",
       "                // almost always do), so we need to clear the canvas so that\n",
       "                // there is no ghosting.\n",
       "                fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n",
       "            }\n",
       "            fig.context.drawImage(fig.imageObj, 0, 0);\n",
       "        };\n",
       "\n",
       "    this.imageObj.onunload = function() {\n",
       "        this.ws.close();\n",
       "    }\n",
       "\n",
       "    this.ws.onmessage = this._make_on_message_function(this);\n",
       "\n",
       "    this.ondownload = ondownload;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_header = function() {\n",
       "    var titlebar = $(\n",
       "        '<div class=\"ui-dialog-titlebar ui-widget-header ui-corner-all ' +\n",
       "        'ui-helper-clearfix\"/>');\n",
       "    var titletext = $(\n",
       "        '<div class=\"ui-dialog-title\" style=\"width: 100%; ' +\n",
       "        'text-align: center; padding: 3px;\"/>');\n",
       "    titlebar.append(titletext)\n",
       "    this.root.append(titlebar);\n",
       "    this.header = titletext[0];\n",
       "}\n",
       "\n",
       "\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n",
       "\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_canvas = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var canvas_div = $('<div/>');\n",
       "\n",
       "    canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n",
       "\n",
       "    function canvas_keyboard_event(event) {\n",
       "        return fig.key_event(event, event['data']);\n",
       "    }\n",
       "\n",
       "    canvas_div.keydown('key_press', canvas_keyboard_event);\n",
       "    canvas_div.keyup('key_release', canvas_keyboard_event);\n",
       "    this.canvas_div = canvas_div\n",
       "    this._canvas_extra_style(canvas_div)\n",
       "    this.root.append(canvas_div);\n",
       "\n",
       "    var canvas = $('<canvas/>');\n",
       "    canvas.addClass('mpl-canvas');\n",
       "    canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n",
       "\n",
       "    this.canvas = canvas[0];\n",
       "    this.context = canvas[0].getContext(\"2d\");\n",
       "\n",
       "    var rubberband = $('<canvas/>');\n",
       "    rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n",
       "\n",
       "    var pass_mouse_events = true;\n",
       "\n",
       "    canvas_div.resizable({\n",
       "        start: function(event, ui) {\n",
       "            pass_mouse_events = false;\n",
       "        },\n",
       "        resize: function(event, ui) {\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
       "        },\n",
       "        stop: function(event, ui) {\n",
       "            pass_mouse_events = true;\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
       "        },\n",
       "    });\n",
       "\n",
       "    function mouse_event_fn(event) {\n",
       "        if (pass_mouse_events)\n",
       "            return fig.mouse_event(event, event['data']);\n",
       "    }\n",
       "\n",
       "    rubberband.mousedown('button_press', mouse_event_fn);\n",
       "    rubberband.mouseup('button_release', mouse_event_fn);\n",
       "    // Throttle sequential mouse events to 1 every 20ms.\n",
       "    rubberband.mousemove('motion_notify', mouse_event_fn);\n",
       "\n",
       "    rubberband.mouseenter('figure_enter', mouse_event_fn);\n",
       "    rubberband.mouseleave('figure_leave', mouse_event_fn);\n",
       "\n",
       "    canvas_div.on(\"wheel\", function (event) {\n",
       "        event = event.originalEvent;\n",
       "        event['data'] = 'scroll'\n",
       "        if (event.deltaY < 0) {\n",
       "            event.step = 1;\n",
       "        } else {\n",
       "            event.step = -1;\n",
       "        }\n",
       "        mouse_event_fn(event);\n",
       "    });\n",
       "\n",
       "    canvas_div.append(canvas);\n",
       "    canvas_div.append(rubberband);\n",
       "\n",
       "    this.rubberband = rubberband;\n",
       "    this.rubberband_canvas = rubberband[0];\n",
       "    this.rubberband_context = rubberband[0].getContext(\"2d\");\n",
       "    this.rubberband_context.strokeStyle = \"#000000\";\n",
       "\n",
       "    this._resize_canvas = function(width, height) {\n",
       "        // Keep the size of the canvas, canvas container, and rubber band\n",
       "        // canvas in synch.\n",
       "        canvas_div.css('width', width)\n",
       "        canvas_div.css('height', height)\n",
       "\n",
       "        canvas.attr('width', width);\n",
       "        canvas.attr('height', height);\n",
       "\n",
       "        rubberband.attr('width', width);\n",
       "        rubberband.attr('height', height);\n",
       "    }\n",
       "\n",
       "    // Set the figure to an initial 600x600px, this will subsequently be updated\n",
       "    // upon first draw.\n",
       "    this._resize_canvas(600, 600);\n",
       "\n",
       "    // Disable right mouse context menu.\n",
       "    $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n",
       "        return false;\n",
       "    });\n",
       "\n",
       "    function set_focus () {\n",
       "        canvas.focus();\n",
       "        canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    window.setTimeout(set_focus, 100);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>')\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items) {\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) {\n",
       "            // put a spacer in here.\n",
       "            continue;\n",
       "        }\n",
       "        var button = $('<button/>');\n",
       "        button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n",
       "                        'ui-button-icon-only');\n",
       "        button.attr('role', 'button');\n",
       "        button.attr('aria-disabled', 'false');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "\n",
       "        var icon_img = $('<span/>');\n",
       "        icon_img.addClass('ui-button-icon-primary ui-icon');\n",
       "        icon_img.addClass(image);\n",
       "        icon_img.addClass('ui-corner-all');\n",
       "\n",
       "        var tooltip_span = $('<span/>');\n",
       "        tooltip_span.addClass('ui-button-text');\n",
       "        tooltip_span.html(tooltip);\n",
       "\n",
       "        button.append(icon_img);\n",
       "        button.append(tooltip_span);\n",
       "\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    var fmt_picker_span = $('<span/>');\n",
       "\n",
       "    var fmt_picker = $('<select/>');\n",
       "    fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n",
       "    fmt_picker_span.append(fmt_picker);\n",
       "    nav_element.append(fmt_picker_span);\n",
       "    this.format_dropdown = fmt_picker[0];\n",
       "\n",
       "    for (var ind in mpl.extensions) {\n",
       "        var fmt = mpl.extensions[ind];\n",
       "        var option = $(\n",
       "            '<option/>', {selected: fmt === mpl.default_extension}).html(fmt);\n",
       "        fmt_picker.append(option)\n",
       "    }\n",
       "\n",
       "    // Add hover states to the ui-buttons\n",
       "    $( \".ui-button\" ).hover(\n",
       "        function() { $(this).addClass(\"ui-state-hover\");},\n",
       "        function() { $(this).removeClass(\"ui-state-hover\");}\n",
       "    );\n",
       "\n",
       "    var status_bar = $('<span class=\"mpl-message\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n",
       "    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n",
       "    // which will in turn request a refresh of the image.\n",
       "    this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_message = function(type, properties) {\n",
       "    properties['type'] = type;\n",
       "    properties['figure_id'] = this.id;\n",
       "    this.ws.send(JSON.stringify(properties));\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_draw_message = function() {\n",
       "    if (!this.waiting) {\n",
       "        this.waiting = true;\n",
       "        this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n",
       "    }\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    var format_dropdown = fig.format_dropdown;\n",
       "    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n",
       "    fig.ondownload(fig, format);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_resize = function(fig, msg) {\n",
       "    var size = msg['size'];\n",
       "    if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n",
       "        fig._resize_canvas(size[0], size[1]);\n",
       "        fig.send_message(\"refresh\", {});\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n",
       "    var x0 = msg['x0'];\n",
       "    var y0 = fig.canvas.height - msg['y0'];\n",
       "    var x1 = msg['x1'];\n",
       "    var y1 = fig.canvas.height - msg['y1'];\n",
       "    x0 = Math.floor(x0) + 0.5;\n",
       "    y0 = Math.floor(y0) + 0.5;\n",
       "    x1 = Math.floor(x1) + 0.5;\n",
       "    y1 = Math.floor(y1) + 0.5;\n",
       "    var min_x = Math.min(x0, x1);\n",
       "    var min_y = Math.min(y0, y1);\n",
       "    var width = Math.abs(x1 - x0);\n",
       "    var height = Math.abs(y1 - y0);\n",
       "\n",
       "    fig.rubberband_context.clearRect(\n",
       "        0, 0, fig.canvas.width, fig.canvas.height);\n",
       "\n",
       "    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n",
       "    // Updates the figure title.\n",
       "    fig.header.textContent = msg['label'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n",
       "    var cursor = msg['cursor'];\n",
       "    switch(cursor)\n",
       "    {\n",
       "    case 0:\n",
       "        cursor = 'pointer';\n",
       "        break;\n",
       "    case 1:\n",
       "        cursor = 'default';\n",
       "        break;\n",
       "    case 2:\n",
       "        cursor = 'crosshair';\n",
       "        break;\n",
       "    case 3:\n",
       "        cursor = 'move';\n",
       "        break;\n",
       "    }\n",
       "    fig.rubberband_canvas.style.cursor = cursor;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_message = function(fig, msg) {\n",
       "    fig.message.textContent = msg['message'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_draw = function(fig, msg) {\n",
       "    // Request the server to send over a new figure.\n",
       "    fig.send_draw_message();\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n",
       "    fig.image_mode = msg['mode'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Called whenever the canvas gets updated.\n",
       "    this.send_message(\"ack\", {});\n",
       "}\n",
       "\n",
       "// A function to construct a web socket function for onmessage handling.\n",
       "// Called in the figure constructor.\n",
       "mpl.figure.prototype._make_on_message_function = function(fig) {\n",
       "    return function socket_on_message(evt) {\n",
       "        if (evt.data instanceof Blob) {\n",
       "            /* FIXME: We get \"Resource interpreted as Image but\n",
       "             * transferred with MIME type text/plain:\" errors on\n",
       "             * Chrome.  But how to set the MIME type?  It doesn't seem\n",
       "             * to be part of the websocket stream */\n",
       "            evt.data.type = \"image/png\";\n",
       "\n",
       "            /* Free the memory for the previous frames */\n",
       "            if (fig.imageObj.src) {\n",
       "                (window.URL || window.webkitURL).revokeObjectURL(\n",
       "                    fig.imageObj.src);\n",
       "            }\n",
       "\n",
       "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
       "                evt.data);\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "        else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n",
       "            fig.imageObj.src = evt.data;\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig[\"handle_\" + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "}\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function(e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e)\n",
       "        e = window.event;\n",
       "    if (e.target)\n",
       "        targ = e.target;\n",
       "    else if (e.srcElement)\n",
       "        targ = e.srcElement;\n",
       "    if (targ.nodeType == 3) // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "\n",
       "    // jQuery normalizes the pageX and pageY\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    // offset() returns the position of the element relative to the document\n",
       "    var x = e.pageX - $(targ).offset().left;\n",
       "    var y = e.pageY - $(targ).offset().top;\n",
       "\n",
       "    return {\"x\": x, \"y\": y};\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys (original) {\n",
       "  return Object.keys(original).reduce(function (obj, key) {\n",
       "    if (typeof original[key] !== 'object')\n",
       "        obj[key] = original[key]\n",
       "    return obj;\n",
       "  }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function(event, name) {\n",
       "    var canvas_pos = mpl.findpos(event)\n",
       "\n",
       "    if (name === 'button_press')\n",
       "    {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x;\n",
       "    var y = canvas_pos.y;\n",
       "\n",
       "    this.send_message(name, {x: x, y: y, button: event.button,\n",
       "                             step: event.step,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "\n",
       "    /* This prevents the web browser from automatically changing to\n",
       "     * the text insertion cursor when the button is pressed.  We want\n",
       "     * to control all of the cursor setting manually through the\n",
       "     * 'cursor' event from matplotlib */\n",
       "    event.preventDefault();\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    // Handle any extra behaviour associated with a key event\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.key_event = function(event, name) {\n",
       "\n",
       "    // Prevent repeat events\n",
       "    if (name == 'key_press')\n",
       "    {\n",
       "        if (event.which === this._key)\n",
       "            return;\n",
       "        else\n",
       "            this._key = event.which;\n",
       "    }\n",
       "    if (name == 'key_release')\n",
       "        this._key = null;\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.which != 17)\n",
       "        value += \"ctrl+\";\n",
       "    if (event.altKey && event.which != 18)\n",
       "        value += \"alt+\";\n",
       "    if (event.shiftKey && event.which != 16)\n",
       "        value += \"shift+\";\n",
       "\n",
       "    value += 'k';\n",
       "    value += event.which.toString();\n",
       "\n",
       "    this._key_event_extra(event, name);\n",
       "\n",
       "    this.send_message(name, {key: value,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n",
       "    if (name == 'download') {\n",
       "        this.handle_save(this, null);\n",
       "    } else {\n",
       "        this.send_message(\"toolbar_button\", {name: name});\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n",
       "    this.message.textContent = tooltip;\n",
       "};\n",
       "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to  previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
       "\n",
       "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n",
       "\n",
       "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n",
       "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
       "    // object with the appropriate methods. Currently this is a non binary\n",
       "    // socket, so there is still some room for performance tuning.\n",
       "    var ws = {};\n",
       "\n",
       "    ws.close = function() {\n",
       "        comm.close()\n",
       "    };\n",
       "    ws.send = function(m) {\n",
       "        //console.log('sending', m);\n",
       "        comm.send(m);\n",
       "    };\n",
       "    // Register the callback with on_msg.\n",
       "    comm.on_msg(function(msg) {\n",
       "        //console.log('receiving', msg['content']['data'], msg);\n",
       "        // Pass the mpl event to the overriden (by mpl) onmessage function.\n",
       "        ws.onmessage(msg['content']['data'])\n",
       "    });\n",
       "    return ws;\n",
       "}\n",
       "\n",
       "mpl.mpl_figure_comm = function(comm, msg) {\n",
       "    // This is the function which gets called when the mpl process\n",
       "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
       "\n",
       "    var id = msg.content.data.id;\n",
       "    // Get hold of the div created by the display call when the Comm\n",
       "    // socket was opened in Python.\n",
       "    var element = $(\"#\" + id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm)\n",
       "\n",
       "    function ondownload(figure, format) {\n",
       "        window.open(figure.imageObj.src);\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy,\n",
       "                           ondownload,\n",
       "                           element.get(0));\n",
       "\n",
       "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
       "    // web socket which is closed, not our websocket->open comm proxy.\n",
       "    ws_proxy.onopen();\n",
       "\n",
       "    fig.parent_element = element.get(0);\n",
       "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
       "    if (!fig.cell_info) {\n",
       "        console.error(\"Failed to find cell for figure\", id, fig);\n",
       "        return;\n",
       "    }\n",
       "\n",
       "    var output_index = fig.cell_info[2]\n",
       "    var cell = fig.cell_info[0];\n",
       "\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function(fig, msg) {\n",
       "    fig.root.unbind('remove')\n",
       "\n",
       "    // Update the output cell to use the data from the current canvas.\n",
       "    fig.push_to_output();\n",
       "    var dataURL = fig.canvas.toDataURL();\n",
       "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
       "    // the notebook keyboard shortcuts fail.\n",
       "    IPython.keyboard_manager.enable()\n",
       "    $(fig.parent_element).html('<img src=\"' + dataURL + '\">');\n",
       "    fig.close_ws(fig, msg);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.close_ws = function(fig, msg){\n",
       "    fig.send_message('closing', msg);\n",
       "    // fig.ws.close()\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
       "    // Turn the data on the canvas into data in the output cell.\n",
       "    var dataURL = this.canvas.toDataURL();\n",
       "    this.cell_info[1]['text/html'] = '<img src=\"' + dataURL + '\">';\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Tell IPython that the notebook contents must change.\n",
       "    IPython.notebook.set_dirty(true);\n",
       "    this.send_message(\"ack\", {});\n",
       "    var fig = this;\n",
       "    // Wait a second, then push the new image to the DOM so\n",
       "    // that it is saved nicely (might be nice to debounce this).\n",
       "    setTimeout(function () { fig.push_to_output() }, 1000);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>')\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items){\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) { continue; };\n",
       "\n",
       "        var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></button>');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    // Add the status bar.\n",
       "    var status_bar = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
       "    var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
       "    button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
       "    button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
       "    buttongrp.append(button);\n",
       "    var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
       "    titlebar.prepend(buttongrp);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(el){\n",
       "    var fig = this\n",
       "    el.on(\"remove\", function(){\n",
       "\tfig.close_ws(fig, {});\n",
       "    });\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(el){\n",
       "    // this is important to make the div 'focusable\n",
       "    el.attr('tabindex', 0)\n",
       "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
       "    // off when our div gets focus\n",
       "\n",
       "    // location in version 3\n",
       "    if (IPython.notebook.keyboard_manager) {\n",
       "        IPython.notebook.keyboard_manager.register_events(el);\n",
       "    }\n",
       "    else {\n",
       "        // location in version 2\n",
       "        IPython.keyboard_manager.register_events(el);\n",
       "    }\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    var manager = IPython.notebook.keyboard_manager;\n",
       "    if (!manager)\n",
       "        manager = IPython.keyboard_manager;\n",
       "\n",
       "    // Check for shift+enter\n",
       "    if (event.shiftKey && event.which == 13) {\n",
       "        this.canvas_div.blur();\n",
       "        // select the cell after this one\n",
       "        var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n",
       "        IPython.notebook.select(index + 1);\n",
       "    }\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    fig.ondownload(fig, null);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.find_output_cell = function(html_output) {\n",
       "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
       "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
       "    // IPython event is triggered only after the cells have been serialised, which for\n",
       "    // our purposes (turning an active figure into a static one), is too late.\n",
       "    var cells = IPython.notebook.get_cells();\n",
       "    var ncells = cells.length;\n",
       "    for (var i=0; i<ncells; i++) {\n",
       "        var cell = cells[i];\n",
       "        if (cell.cell_type === 'code'){\n",
       "            for (var j=0; j<cell.output_area.outputs.length; j++) {\n",
       "                var data = cell.output_area.outputs[j];\n",
       "                if (data.data) {\n",
       "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
       "                    data = data.data;\n",
       "                }\n",
       "                if (data['text/html'] == html_output) {\n",
       "                    return [cell, data, j];\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "// Register the function which deals with the matplotlib target/channel.\n",
       "// The kernel may be null if the page has been refreshed.\n",
       "if (IPython.notebook.kernel != null) {\n",
       "    IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
       "}\n"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<img src=\"\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# run 2D PSV code\n",
    "vy, vy = psv_mod(nt, nx, ny, dt, dh, rho, lam, mu, clip, isnap, X, Y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The vy wavefield snapshot shows the direct P- and S-wave propagating in the homogeneous block model. Note the amplitude distribution along the circumference of the wavefront. Due to the impact source, maximum amplitudes of the P-wave occur along the direction of the impact and perpendicular to the impact for the S-wave. Amplitude minima are located at a 90° offset.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---\n",
    "__Text provided under a Creative Commons Attribution license, CC-BY.  All code is made available under the GNU GPL 3.0 license.  (c) Daniel Köhn, Denise De Nil 2017. This notebook is inspired by the [CFDPython](https://github.com/barbagroup/CFDPython) course (c) Lorena A. Barba, Gilbert F. Forsyth 2015__"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [default]",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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